2009/02/17

Asymptotic Behavior of Solutions to Linear Equations

Apart from the interesting article in the last post, Justin also sent me the following interesting question:

Consider

– – – – – – (E)

where the constant is positive and is continuous on [).

(a) Show that the general solution to equation (E) can be written in the form

where is a non-negative constant.

(b) If for , where and are non-negative constants, show that

for .

(c) Let satisfy the same equation as (E) but with forcing function . That is

,

where is continuous on [). Show that if

for ,

then

for .

(d) Show that if as , then any solution of (E) satisfies as .
[Hint: Take and in part (c).]

(e) Suppose brine solution containing kg of salt per litre at time runs into a tank of water at a fixed rate and that the mixture, kept uniform by stirring, flows out at the same rate. Given that as , by using the result in part (d), determine the limiting concentration of the salt in the tank as .

I think the question is well set and suitable for self-study at secondary school standard, try.